Integral closure

Let $S$ be a ring and $R$ a subring of $S$. We say that an element $s\in S$ is integral over $R$ if there is a monic polynomial $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$, with each $a_i\in R$ so that $f(s)=0$. The elements of $S$ that are integral over $R$ form a subring $T$ of $S$ which contains $R$. We call $T$ the integral closure of $R$ in $S$. If $T=S$, then we say that $S$ is integral over $R$. If $T=R$, then we say that $R$ is integrally closed in $S$. If $R$ is integrally closed in its field of fractions, then we call it integrally closed.

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